Thermally driven motion of strongly heated fluids

Approximate equations which are derived for the thermally driven motion of strongly heated, shallow fluids were examined. The equations follow from formal perturbation theory, using the nondimensionalized depth epsilon of the fluid as the smaller parameter. They govern the zeroth order temperature, pressure, and density. The method used is also applied to a weakly heated, deep fluid, using the nondimensionalized strength delta' of the heat source distribution as the small parameter. In the later case, the equations govern the first order perturbations in temperature, pressure, and density, while the unperturbed situation is the so called adiabatic state. The Boussinesq equations are recovered for a weakly heated, shallow fluid, independent of the relative magnitude of delta' and epsilon. For the case of a weakly heated, shallow ideal gas maintained at constant volume and heated throughout the volume, the equations account for the effect of perturbation pressure on perturbation density and on compression heating. A final case considered is that of a strongly heated, deep liquid with a small coefficient of thermal expansion. The energy equation for this case is found to be significantly different from that for a weakly heated, deep liquid. For two dimensional geometries, the various sets of equations derived are amendable to a common numerical method of solution.